Is there a way to get continuous radial/polar coordinates?
You could avoid doing the trigonometric stuff the Gradient &gt; Radial node has already done for you, by, instead of avoiding the discontinuity at 0, using it as the start of your segment:

At its most primitive, the start-angle is measured out of 1, clockwise from 9 o'clock, and the segment is anticlockwise from there.This set-up works round the clock, through negative and &gt;1 start-angles.
But I guess the disadvantage is the amount of plumbing you have to do to wrangle the interface into offering you center of arc, angle of arc, in degrees, clockwise from 12 o'clock:

(Maybe there are ways of cutting this down?) ... anyway, that's quite a few nodes to avoid one trig. calculation per shading point. Given modern kit, this is probably old-fashioned :)
Edit:
It may be better to forget the Radial altogether and roll-your-own:

.. which is added to the .blend: